Randomstrasse101 Open Problems (2024) — Conjecture 4 (Density threshold for Global Synchrony)

Definitions and mathematical statement

Let $V(G)$ denote the vertex set of a finite graph $G$, and write \[ \delta(G):=\min_{v\in V(G)}\deg_G(v) \] for its minimum degree, with the graph and degree conventions understood as intended by the source. The supplied record does not specify those conventions or define the property "globally synchronizing." Consequently, that property is retained as an unresolved source predicate; a self-contained dynamical definition cannot be recovered from the supplied evidence.

The supplied conjecture is: \[ \forall\varepsilon\in{\mathbb{R}}_{>0}\quad \exists n\in{\mathbb{Z}}_{\geq 1}\quad \exists G\quad \left[ |V(G)|=n\ \land\ \delta(G)\geq\left(\frac34-\varepsilon\right)n \ \land\ G\text{ is not globally synchronizing} \right]. \] Here $G$ must belong to the graph class intended by the source, which is not specified in the supplied record. The integer $n$ and graph $G$ may depend on $\varepsilon$. The statement asserts existence of a positive integer $n$ for each $\varepsilon>0$; it does not explicitly require arbitrarily large $n$.

Short English statement

For every $\varepsilon>0$, the conjecture asks for a graph on some positive number $n$ of vertices with minimum degree at least $(3/4-\varepsilon)n$ that is not globally synchronizing. The intended graph conventions and the definition of global synchrony are missing, so the formulation requires clarification and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $800839$, exact source problem code catalogue entry; supplied title: "Randomstrasse101 Open Problems (2024) --- Conjecture 4 (Density threshold for Global Synchrony)." The registry attributes the problem to Bandeira et al., gives proposed year 2025, and labels it open. This status is supplied metadata, not an independently verified mathematical assessment.
• Supplied source reference: Bandeira et al., Randomstrasse101: Open Problems of 2024 (2025), Conjecture 4, https://arxiv.org/abs/2504.20539. The supplied background identifies the context as "Density threshold for Global Synchrony" and the statement describes an extremal combinatorics version of the global synchrony question.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims; no source visit or independent verification is asserted here. Registry upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.